Tukey lambda distribution

Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly.

Tukey lambda distribution

Probability density function
Notation	Tukey(λ)
Parameters	λ ∈ ℝ — shape parameter
Support	x ∈ [ − 1 /λ,  1 /λ ]    if    λ > 0 ,   x ∈ ℝ           if    λ ≤ 0 .
PDF	${\displaystyle \left(\ Q(\ p\$ ;\lambda \ ),\ {\frac {1}{\ q(\ p\ ;\lambda \ )\ }}\ \right)\quad ~{\mathsf {for\ any}}~\quad p\;:\;0\leq \ p\ \leq \ 1}
CDF	${\displaystyle {\Bigl (}\ Q(p;\lambda ),\ p\ {\Bigr )}~~{\mathsf {for\ any}}~~p\;:\;0\leq \ p\ \leq \ 1~}$   (general case)   ${\displaystyle {\frac {1}{\ e^{-x}+1\ }}\quad ~{\mathsf {if}}~\quad \lambda \ =\ 0\quad }$ (special case exact solution)
Mean	${\displaystyle 0\quad ~{\mathsf {if}}~\quad \lambda >-1\ }$
Median	0
Mode	0
Variance	${\displaystyle {\frac {2}{\ \lambda ^{2}\ }}\left(\ {\frac {1}{\ 1+2\ \lambda \ }}-{\frac {\ \Gamma \,\!(\lambda +1)^{2}\ }{\ \Gamma \,\!(\ 2\ \lambda +2\ )\ }}\right)\quad ~{\mathsf {if}}~\quad \lambda >-{\tfrac {\ 1\ }{2}}\ }$   ${\displaystyle {\frac {\ \pi ^{2}\ }{3}}\qquad \qquad ~{\mathsf {if}}~\quad \lambda \ =\ 0}$
Skewness	${\displaystyle 0\qquad \qquad ~{\mathsf {if}}~\quad \lambda >-{\tfrac {\ 1\ }{3}}\ }$
Ex. kurtosis	${\displaystyle ~{\frac {\ (2\ \lambda +1)^{2}\cdot g_{2}^{2}\cdot {\big (}\ 3\ g_{2}^{2}-4\ g_{1}\ g_{3}+g_{4}\ {\big )}\ }{\ (\ 8\ \lambda +2\ )\cdot g_{4}\cdot {\big (}\ g_{1}^{2}-g_{2}\ {\big )}^{2}}}\ -\ 3\quad {\mathsf {if}}~~\lambda >0\ ,}$   ${\displaystyle {\frac {\ 6\ }{5}}\qquad \qquad ~{\mathsf {if}}~\quad \lambda \ =\ 0\$ ;}   ${\displaystyle {\mathsf {where}}\quad g_{k}\ \equiv \ \Gamma \,\!(\ k\,\lambda +1\ )\quad {\mathsf {and}}\quad \lambda \ >-{\tfrac {\ 1\ }{4}}~.}$
Entropy	${\displaystyle h(\lambda )=\int _{0}^{1}\ln {\bigl (}\ q(p;\lambda )\ {\bigr )}\ \operatorname {d} p~}$
CF	${\displaystyle \phi (t;\lambda )=\int _{0}^{1}\exp {\bigl (}\ i\ t\ Q(p;\lambda )\ {\bigr )}\ \operatorname {d} p~}$

The Tukey lambda distribution has a single shape parameter, λ, and as with other probability distributions, it can be transformed with a location parameter, μ, and a scale parameter, σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.